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arXiv:2308.11823 · Physical review. B./Physical review. B

Solving Fermi-Hubbard-type models by tensor representations of backflow corrections

Yu-Tong Zhou, Zheng-Wei Zhou, Xiao Liang

Published Jun 4, 2024• 2 citations
arXivHTMLPDF

Abstract

The quantum many-body problem is an important topic in condensed matter physics. To efficiently solve the problem, several methods have been developed to improve the representation ability of wave functions. For the Fermi-Hubbard model under periodic boundary conditions, current state-of-the-art methods are neural network backflows and the hidden fermion Slater determinant. The backflow correction is an efficient way to improve the Slater determinant of free particles. In this work we propose a tensor representation of the backflow-corrected wave function; we show that for the spinless $t\text{\ensuremath{-}}V$ model, the energy precision is competitive or even lower than current state-of-the-art fermionic tensor network methods. For models with spin, we further improve the representation ability by considering backflows on fictitious particles with different spins, thus naturally introducing nonzero backflow corrections when the orbital and the particle have opposite spins. We benchmark our method on molecules in the STO-3G basis and the Fermi-Hubbard model with periodic and cylindrical boundary conditions. We show that the tensor representation of backflow corrections achieves competitive or even lower-energy results than current state-of-the-art neural network methods.

DOI
10.1103/physrevb.109.245107

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The quantum many-body problem is an important topic in condensed matter physics. To efficiently solve the problem, several methods have been developed to improve the representation ability of wave functions. For the Fermi-Hubbard model under periodic boundary conditions, current state-of-the-art methods are neural network backflows and the hidden fermion Slater determinant. The backflow correction is an efficient way to improve the Slater determinant of free particles. In this work we propose a tensor representation of the backflow-corrected wave function; we show that for the spinless t\ensuremath-Vt\text{\ensuremath{-}}Vt\ensuremath-V model, the energy precision is competitive or even lower than current state-of-the-art fermionic tensor network methods. For models with spin, we further improve the representation ability by considering backflows on fictitious particles with different spins, thus naturally introducing nonzero backflow corrections when the orbital and the particle have opposite spins. We benchmark our method on molecules in the STO-3G basis and the Fermi-Hubbard model with periodic and cylindrical boundary conditions. We show that the tensor representation of backflow corrections achieves competitive or even lower-energy results than current state-of-the-art neural network methods.
We apply the Tensor-Backflow method to investigate the Fermi-Hubbard model on two-dimensional lattices up to 256 sites, exploring various interaction strengths UUU, electron fillings nnn, next-nearest-neighbor hopping t′t't′, and boundary conditions. By considering backflow terms from nearest- or next-nearest-neighbor sites, we achieve competitive results without enforcing geometric symmetries on the variational wave-function. The optimizations were stable from a prior unrestrictied Hartree-Fock state, followed by adding backflow corrections. Meanwhile, changing interaction strengths in the prior unrestrictied Hartree-Fock state is helpful to bypass the local minima. When t′t't′=0, by considering nearest-neighbor backflow terms, linear stripe order emerges successfully for the case of nnn=0.875 and UUU=8 on a 16×1616 \times 1616×16 lattice with periodic boundary conditions. In a similar case with open boundary conditions, the energy obtained is only 4.5×10−44.5 \times 10^{-4}4.5×10−4 higher than the state-of-the-art method fPEPS with bond dimension DDD=20. Compared to state-of-the-art neural network methods, the energies obtained using the Tensor-Backflow approach are competitive, with relative errors below 5×10−35 \times 10^{-3}5×10−3. For nnn=0.8 and nnn=0.9375, direct optimizations yield results consistent with the phase diagram from AFQMC. When t′t't′=-0.2, considering next-nearest-neighbor backflow terms leads to energies that are either competitive with or even lower than those from state-of-the-art neural network approaches. For instance, for nnn=0.875 and UUU=8 on a 12×1212 \times 1212×12 lattice with periodic boundary conditions, the energy obtained is 8.1×10−48.1 \times 10^{-4}8.1×10−4 lower than that from the neural network result. Thus, the Tensor-Backflow method demonstrates strong representational capabilities for solving the Fermi-Hubbard model.
Abstract Strongly correlated electrons give rise to an array of electronic properties increasingly exploited in many emerging materials and molecular processes. However, the reliable numerical simulation of this quantum many-body problem still poses an outstanding challenge, in particular when accounting for the fermionic statistics of electrons. In this work, we introduce a compact and systematically improvable fermionic wave function based on a CANDECOMP/PARAFAC (CP) tensor decomposition of backflow correlations in second quantization. This ansatz naturally encodes many-electron correlations without the ordering dependence of other tensor decompositions. We benchmark its performance against standard models, demonstrating improved accuracy over comparable methods in Fermi-Hubbard and molecular systems and competitive results with state-of-the-art density matrix renormalization group (DMRG) in ab initio 2D hydrogenic lattices. By considering controllable truncations in the rank and range of the backflow correlations, as well as screening the local energy contributions for realistic Coulomb interactions, we obtain a scalable and interpretable approach to strongly correlated electronic structure problems that bridges tensor factorizations and machine learning-based representations.
Among the variational wave functions for fermionic Hamiltonians, neural network backflow (NNBF) and hidden fermion determinant states (HFDS) are two prominent classes that provide accurate approximations to the ground state. Here we develop a unifying view of fermionic neural quantum states casting them all in the framework of NNBF. NNBF wave functions have configuration-dependent single-particle orbitals (SPO) which are parameterized by a neural network. We show that HFDS can be written as a NNBF wave function with a restricted low-rank rrr additive correction to the SPO times a neural-network generated determinant of an $r\ifmmode\times\else\texttimes\fi{}r$ matrix. Furthermore, we show that in NNBF wave functions, this $r\ifmmode\times\else\texttimes\fi{}r$ determinant can generically be removed when rrr is less than or equal to the number of fermions, at the cost of further complicating the additive SPO correction increasing its rank by rrr. We numerically and analytically compare additive SPO corrections generated by the product of two matrices with inner dimension rrr. We find that larger rrr wave functions span a larger space and give evidence that simpler and more direct updates to the SPO's tend to be more expressive and better energetically. These suggest the standard NNBF approach is preferred amongst other related choices. Finally, we uncover that the row selection used to select single-particle orbitals allows significant sign and amplitude modulation between nearby configurations and is partially responsible for the quality of NNBF and HFDS wave functions.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O(N^3)O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N2)O(N^2)O(N2) or less. We show that the accuracy of the approximation can be improved by optimizing the variance of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
We describe a class of neuralized fermionic tensor network states (NN-fTNS) that introduce non-linearity into fermionic tensor networks through configuration-dependent neural network transformations of the local tensors. The construction uses the fTNS algebra to implement a natural fermionic sign structure and is compatible with standard tensor network algorithms, but gains enhanced expressivity through the neural network parametrization. Using the 1D and 2D Fermi-Hubbard models as benchmarks, we demonstrate that NN-fTNS achieve order of magnitude improvements in the ground-state energy compared to pure fTNS with the same bond dimension, and can be systematically improved through both the tensor network bond dimension and the neural network parametrization. Compared to existing fermionic neural quantum states (NQS) based on Slater determinants and Pfaffians, NN-fTNS offer a physically motivated alternative fermionic structure. Furthermore, compared to such states, NN-fTNS naturally exhibit improved computational scaling and we demonstrate a construction that achieves linear scaling with the lattice size.
Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this Letter, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman and Cohen [Phys. Rev. 102, 1189 (1956)10.1103/PhysRev.102.1189], adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to learn the optimal transformation via variational Monte Carlo calculations. NNB directly dresses a mean-field state, can be systematically improved, and directly alters the sign structure of the wave function. It generalizes the standard backflow [L. F. Tocchio et al., Phys. Rev. B 78, 041101(R) (2008)10.1103/PhysRevB.78.041101], which we show how to explicitly represent as a NNB. We benchmark the NNB on Hubbard models at intermediate doping, finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.